By B. Stenstrom

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97 Introduction In various branches of mathematics and its applications there arises a need to use representations, and so problems of their classiﬁcation become urgent; cf. [18, 20, 32, 46, 47, 50, 54, 65, 66, 86]. If one takes into account that a representation is a two-sorted algebraic system (a pair) then the systematics of representations is facilitated. The book [35] is 20 C HAPTER I. REPRESENTATIONS OF SEMIGROUPS AND ALGEBRAS written from this point of view, and, furthermore, there is visible evidence of this in the note [57] and in the survey [41].

One consequence of this fact deserves special attention because of an application in the last section. 1. Let A be an arbitrary (additively written) Abelian group, B an arbitrary group, and E the unit group. The acting group of the pair (A, E) (ZB, B) is isomorphic to AwrB. P ROOF. The proof amounts to applying the preceding formula to the pairs (A, E) and (ZB, B). Let us also give a sketch of the proof of formula (3), because of the lack of a suitable reference. 3. Triangular products and stability of representations 25 B2 2 Let there be given an arbitrary pair (A1 , B1 ).

68 3. Powers of the fundamental ideal and stability of representations of groups and semigroups . . . . . . . . . . . . . . . . . . . 1. Preliminary topics; on the terminal of nilpotent groups . . . . . . . 2. Construction of stable representations of groups with the aid of the triangular product . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Generalized measure subgroups of ﬁnite groups . . . . . . . . . . 4. Mal’cev nilpotency and stability of semigroups .